Derive the expression for the speed of sound in an ideal gas assuming adiabatic…

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

Derive the expression for the speed of sound in an ideal gas assuming adiabatic propagation of the disturbance. (i) Starting from the relation

pV^{\gamma}=\text{constant}

for an adiabatic process, show that the effective bulk modulus of the gas for a small disturbance is

B = \gamma p

. (ii) Hence obtain the expression

v = \sqrt{\gamma p / \rho}

for the speed of sound in the gas. (iii) Newton’s result is

v_N = \sqrt{p/\rho}

. Show that the ratio

v / v_N = \sqrt{\gamma}

and evaluate this ratio for air taking

\gamma = 1.40